4.2 The Venables model, and an extension with dis- crete sectorscrete sectors
4.2.3 A model with discrete sectors
With this model, we will try to get some insight into the different types of equilibrium that obtain when we vary the IO matrix. In order to limit the possible number of equilibria and keep the analysis manageable, we only look at the simplest possible setup: a situation where there are two indus- trial sectors and two regions. This will allow us to present the results in a graphical manner later on. An extension to more sectors is straightfor- ward and pursued in chapter 15 of Fujita et al. (1999), among others. We have made a different generalization in chapter 3, where we discussed a continuous IO structure.
We assume two regions and two kinds of firms, agricultural and indus- trial. The industrial firms are divided into two sectors. The products of all the different firms are consumed in both regions by agents who maximize utility,
U =A1−à1−à2Qà11Qà22 (4.10) whereAis consumption of the agricultural good andQiis the consumption of products of (industrial) sectori (i = 1,2). As before, we assume that agricultural good is homogeneous and freely tradeable across regions. It will serve as the num´eraire. As follows from (4.10), the agricultural sector receives a fixed fraction1−à1−à2of each agent’s income.
The industrial goods are heterogenous again, and each sector is subject
to monopolistic competition. The aggregation goes according to Qi=
nq
σ−1 σ
i +n∗(qi∗)σ−1σ σ−1σ
(4.11) for i = 1,2. This equation is similar to (4.4), but it now operates on the sectoral level. Notice that we assume that the values of σ are equal for the two sectors. This is not essential to the results, but does simplify the analysis considerably. Notice also that, compared to (4.4), we have left out the scaling parameterψ. This parameter, which is meant to vary the level of costs of intermediate goods, will be reintroduced at the appropriate level below.
The differentQi’s are themselves aggregated by consumers (as in for- mula 4.10 above) and serve as bundles of intermediate products. The ag- gregation ofQi’s into a factor of production is sector-specific and follows from each sector’s production function. This function is
yi= 1 φãθαLαi
1
ψiãθη,iQηi,1iQ1−ηi,2 i 1−α
−F. (4.12)
This function is similar to (4.2):yi is the production of a firm in sectori.9 It uses labor and bundles of intermediate productQi,k withk= 1,2the sup- plying sector. The bundles are from formula (4.11); we again assume that the elasticity of substitution between products of different producers in the same sector,σ, is the same for final and intermediate demand. Knowing this, when it comes to pricesetting the producer does not have to worry about the different clients and can set the same price for all: the usual markup over marginal costs, MCãσ/(σ −1). Finally, each firm faces a fixed costF that is paid in the final product.
There are a number of constants in formula (4.12). Theθ’s are defined as follows:
θα = α−αã(1−α)α−1, θη,i = ηi−ηiã(1−ηi)ηi−1,
and serve to normalize associated costs. The marginal costs for a firm in sectoriin thehomeregion are
M Ci = φãwαã( ˜Gi)1−α, (4.13) G˜i = ψiãGη1iãG1−η2 i, (4.14) Gi = np1−σi +n∗
p∗i τ
1−σ!1−σ1
. (4.15)
9Notice that we do not index by region—it is assumed that the production functions are similar in both regions.
These expressions follow directly from (4.12). Note thatGi is the price in- dex of goods from sectori in the home region, andG˜i is the price index of intermediate goods, used by sectori, in thehomeregion. For the foreign region,G∗i andG˜∗i could be defined.
From (4.13)-(4.15), we see that marginal costs are a weighted geometri- cal average of wage costs in thehomeregion and the price of intermediate goods. The latter consist in turn of a weighted average of the price indices of goods from sectors1 and2. The price index of goods from sectori, fi- nally, is a weighted average of all prices in the sector, both of firms in the homeregion and in theforeignregion. Note that in order to use prices from the other region, we have to take the transport costs into account.
From the coefficientsηiin formula (4.12)we can construct a two-by-two IO matrix,
IO=
η1 η2
1−η1 1−η2
. (4.16)
We defined an IO matrix as containing the shares of the budget for interme- diates that go to the different sectors. The columns sum to one, indicating that the total budget for intermediates is exhausted. This matrix can be constructed from an IO table by dividing the entries (the flow of trade from one sector to another) by their column sums. Thus, for instance,η1 is the share of their budget for intermediate goods that firms in sector1spend on products from their own sector.
So far, we have left the scaling constantsφandψunspecified. As before, we will set φ = 1/θα for a simplification of (4.12). For the same reason, we could setψi = 1/θη,i for all ifor a baseline result and compare cases where different values ofψlead to a different outcome. However, another issue comes to the front, which is the result of our assumption that discrete sectors are aggregated using a Cobb-Douglas function (as in 4.10). First of all, we need to recognize that the precise categorization introduced into the model is often the result of a judgement call, affected by factors such as data availability. Depending on data, we could carve the economy into two sectors or into twenty. If we want to compare the results of the two-sector model to those in a twenty-sector model, they should at least be the same for some special twenty-sector cases.
A natural point of departure is a generalization of the one-sector model of section 4.2 into ans-sector model (s≥2) with the IO matrixηi,j = 1/sfor alli, j. In this case, intermediate bundles consist of equal amounts of prod- ucts fromeachproducer, regardless of the number of sectors. It seems nat- ural that the price index of intermediatesG˜ (which is equal across sectors) should be the same for any value of s. Whether the intermediate goods come from two sectors or from twenty seems more of an administrative concern than something that would influence the price of that bundle.
In order to achieve such equivalence, it is important to remember that
the monopolistic competition that exists between firms does not cross the sector boundary. Firms within the same sector are monopolistic competi- tors, but each sector in total is guaranteed a fixed share of each budget spent on industrial goods. The latter is a result of our various Cobb-Douglas as- sumptions. Therefore, the love-of-variety effect that causes increasing re- turns to the number of firms, only works on the number of firmswithin a sector. In this model, carving the economy up into many sectors thus has the undesirable effect of reducing overall efficiency.10
This can be seen as follows: suppose we convert a single-sector econ- omy into ans-sector economy, with all entries of the IO matrix equal to1/s.
The price index of intermediate goods in each case is G˜ =
nãp1−σ1−σ1
= n1−σ1 ãp (4.17)
G˜s = hn
s ãp1−σ i1−σ1
= G˜ã 1
s 1−σ1
(4.18) Here,G˜ is the one-sector price index andG˜s is the s-sector price index of a bundle of intermediates. Assis larger than one,G <˜ G˜s, or, the choice of the number of sectorss affects the price of intermediates. In order to preclude this result we introduces, the number of sectors, intoψi: below, we set
ψi = s1−σ1 θη,i
. (4.19)
This way, results between different categorizations are comparable in prin- ciple.
The model is now operational. We can compute the equilibrium for a re- gion, given the number of firms and the prices of goods in the other region, and given the demand from the other region for home products. A detailed description of the solution method is provided in the next paragraph.